The history of set theory is rather different from the
history of most other areas of mathematics. For most areas a long process can
usually be traced in which ideas evolve until an ultimate flash of inspiration,
often by a number of mathematicians almost simultaneously, produces a discovery
of major importance.

Set theory however is rather different. It is
the creation of one person, Georg
Cantor. Before we take up the main story of Cantor's
development of the theory, we first examine some early contributions.

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Welcome to the set tutorial.
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Below you will see a list of this unit's page headings and the subjects
discussed in each of these pages.
Click on any one of them in order to get started.

Subjects to be Learned

Contents

Definition (Equality of sets): Two sets are equal
if and only if they have the same elements.
More formally, for any sets A and B,
A = B if and only if x [ xAxB ] .

Thus for example {1, 2, 3} = {3, 2, 1} , that is the order
of elements does not matter, and {1, 2, 3} = {3, 2, 1, 1},
that is duplications do not make any difference for sets.

Definition (Subset): A set A is a
subset of a set B if and only if everything in A
is also in B.
More formally, for any sets A and B,
A is a subset of B,
and denoted by AB, if and only if x [ xAxB ] .
If AB, and AB, then A is said to be a proper
subset of B and it is denoted by AB .

For example {1, 2}
{3, 2, 1} .
Also {1, 2}
{3, 2, 1} .

Definition(Cardinality): If a set S has n
distinct elements for some natural number n, n is
the cardinality (size) of S and S
is a finite set. The cardinality of S
is denoted by |S|.

For example the cardinality of the set {3, 1, 2} is 3.

Definition(Empty set): A set which has no elements
is called an empty set.
More formally, an empty set, denoted
by ,
is a set that satisfies the following:xx
,
where
means "is not in" or "is not a member of".

Note that
and {}
are different sets. {}
has one element namely
in it. So {}
is not empty. But
has nothing in it.

Definition(Universal set): A set which has all the
elements in the universe of discourse is called a universal set.
More formally, a universal set,
denoted by U , is a set that satisfies the following:xx
U .

Three subset relationships involving empty set and universal set are listed
below as theorems without proof. Their proofs are found
elsewhere.

Note that the set A in the next four theorems are
arbitrary. So A can be an empty set or universal set.

Theorem 1: For an arbitrary set AAU .

Theorem 2: For an arbitrary set AA .

Theorem 3: For an arbitrary set AAA .

Definition(Power set): The set of all subsets of a set
A is called the power set of A
and denoted by 2A or (A)
.